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Since we know that $E|Z|_\infty\sim \sqrt{\log n}$, the first easy case is if $|x|_\infty=o( \sqrt{\log n})$. In that case, $E |x+z|_\infty \sim E|z|_\infty$ in the sense that the ratio goes to $1$ a …

This question can be cast in the language of branching random walks: at each (discrete) time, a particle splits with probability $p$ and each of the children moves $\log \epsilon$; if it did not split …

I don't know how to give a formula for the mean. However the following gives an approach that is often useful asymptotically. Its success depends on the relation between $n$, $N$, $j$, which you did n …

The expected exit time (from a ball of radius 1) is $(1+o_\delta(1))/\delta^2$, regardless of the choice of strategy (the $o_\delta(1)$ term does depend on the strategy). Indeed, write $X_n=\sum_{i=1} …

This is a partial answer, in the regime that the probability in question is close to $1$. I normalize so that $EY_i=1$. The example $a_i=1/\sqrt{n}$ shows that you need to take $t\geq \sqrt{n}/2+O(1) …

See Maximal Arithmetic Progressions in Random Subsets by Benjamini, Yadin and Zeitouni, ECP 12: 365-376 (2007) and the erratum in ECP 17: 1-1 (2012). See also the extensions in M.-Z. Zhao and H.-Z. Zh …

I believe you are missing an $n$ in your definition of $K_n(t)$, that is $K_n(t)=\log E(e^{tnS_n})$. I assume in the sequel that this is what you meant. If $S_n$ satisfies the large deviations princip …

Your field is the $2$-spin, that is can be represented as $Z_x=\sum_{i,j} J_{ij} x_i x_j$, where $J_{i,j} $ are iid Gaussian (up to symmetry, ie $J_{i,j}=J_{j,i}$ and the diagonal has twice the varian …

This edit reflects the actual question asked, and corrects an earlier answer. You can rewrite the process $\int_0^t \sigma_s dW_s$ as a time change of Brownian motion, where the time change is given b …

The minimum (which is an infimum) is $-\infty$. We have that $\limsup_{t\to 0} B_t/\sqrt{t}=\infty$ (this follows in particular from the LIL). This means that there exists $t_M\in (0,1)$ arbitrarily s …

First, there is a typo there - should be E instead of P, as in the second display in your question. Second, the argument as written is not quite right, but it can be rescued. Indeed, $\sup_{t<\delta} …

"Cover times for Brownian motion and random walks in two dimensions": Annals Math 160 (2004), 433-464, By Dembo, Peres, Rosen, Zeitouni. See Theorem 1.4

There is a small literature on these topics, mostly from the 90s. The names to look for are A. Jakubowski and M. Kobus (alone and together). For an example see Theorem 1.2 in1 https://www.sciencedirec …

I will consider stationary Gaussian processes $X_v$ indexed by $v\in Z^d$, not continuous time (the argument for continuous time requires a bit of extra work, and some assumptions on the short-time be …

For standard Gaussians, and with the matrix $W/n$, the proof of the LDP given by Ben Arous-Guionnet adapts to the Wishart setup. However, you will have different scalings and so the non-commutative en …